Minimum Allocation Probabilities and Exploration Floors

Why Put a Floor on Assignment Probabilities?

A DBCD uses response estimates and the current allocation to adapt the chance of assigning the next participant. When one arm's resulting probability becomes very small, that arm may receive few additional participants, even if it remains part of the design. A specified probability floor prevents the next-participant assignment chance from falling below a chosen minimum. The floor is an operational safeguard on the randomization step, not a redefinition of the target allocation.

To state the distinction precisely, let \(q_i(n)\) be the target share for arm \(i\) at step \(n+1\), where \(q_i(n)\geq 0\) and \(\sum_{i=1}^K q_i(n)=1\). Let \(N_i(n)\) be the number assigned to arm \(i\) among the first \(n\) participants, and let \(\hat\pi_i(n)=N_i(n)/n\) when \(n>0\). For positive target shares and positive current allocation proportions, the usual multi-arm DBCD raw weights and raw assignment probabilities can be written as

Here \(\gamma\geq 0\) is the DBCD tuning parameter, and \(r_i(n)\) is the probability before applying a floor. Initialization is needed when allocation proportions or response estimates are not yet available; the formulas below apply once the raw probability vector has been defined. In what follows, suppress the step index and write \(r_i\) for the raw probabilities. They satisfy \(r_i>0\) and \(\sum_i r_i=1\).

Definition and Feasibility

The feasibility condition follows because \(K\) probabilities each at least \(\varepsilon\) must sum to at least \(K\varepsilon\). Thus \(K\varepsilon\leq1\) is necessary. It is sufficient: the uniform vector \(p_i=1/K\) satisfies the constraint whenever \(\varepsilon\leq1/K\). If \(\varepsilon=1/K\), the only feasible vector is uniform; if \(\varepsilon=0\), the constraint imposes no positive lower bound.

For \(0\leq\varepsilon<1/K\), the floor-and-rescale transformation has the form

where \(c>0\) is chosen to make the probabilities sum to one. Arms for which \(c r_i\leq\varepsilon\) are pinned at the floor. The other arms retain their relative raw probabilities, since for any two unpinned arms \(i\) and \(j\), \(p_i/p_j=r_i/r_j\). At a tie, where \(c r_i=\varepsilon\), either description gives the same probability.

If \(F\) is the set of pinned arms and \(B\) is the set of unpinned arms, then, provided \(B\) is nonempty,

The classification must be consistent: \(c r_i\leq\varepsilon\) for \(i\in F\), and \(c r_i\geq\varepsilon\) for \(i\in B\). The formula for \(c\) cannot be used when \(B\) is empty because its denominator would be zero. In the endpoint case \(\varepsilon=1/K\), set \(p_i=\varepsilon=1/K\) for every arm. This is the uniform vector and is the only feasible assignment vector; no rescaling calculation is needed.

Why the Rescaling Rule Works

For a proposed pinned set \(F\), the pinned probabilities use total mass \(|F|\varepsilon\). The remaining arms must therefore receive mass \(1-|F|\varepsilon\). Assigning that mass in proportion to their raw probabilities gives \(p_i=c r_i\) for \(i\in B\). Summing those probabilities determines the multiplier:

To find a consistent set, begin with \(F\) empty, calculate \(c=1/\sum_i r_i=1\), and pin any arm with \(c r_i<\varepsilon\). Recalculate \(c\) using the remaining arms, and repeat if any remaining rescaled probability is below the floor. With positive \(r_i\) and \(\varepsilon<1/K\), a consistent set has at least one unpinned arm: if every arm were pinned, their probabilities would sum to \(K\varepsilon<1\), contradicting the requirement that they sum to one. When \(\varepsilon=1/K\), use the endpoint rule instead.

The transformation is also uniquely specified by its common rescaling factor. The function \(g(c)=\sum_i\max\{\varepsilon,c r_i\}\) is continuous and nondecreasing. For \(\varepsilon<1/K\), it begins at \(K\varepsilon<1\) for sufficiently small positive \(c\), and grows without bound as \(c\) increases because every \(r_i>0\). Once it exceeds its initial flat portion, it is strictly increasing: at least one term with positive \(r_i\) is increasing. Consequently, exactly one \(c\) solves \(g(c)=1\). This verifies that the floor rule is well-defined under the stated conditions.

Keep the Floor Separate From the Target

A target allocation and an assignment probability answer different questions. The target \(q_i(n)\) describes the allocation share the design is tracking at a particular step. The raw DBCD probability \(r_i(n)\) is calculated from that target, the current allocation, and \(\gamma\). The floored probability \(p_i(n)\) is the final chance used to randomize the next participant. A constrained target, by contrast, changes the target shares supplied to the allocation rule. These operations can lead to different probability vectors and have different interpretations.

One alternative is to change the target shares before calculating the DBCD probabilities, for example by imposing constraints on \(q_i(n)\). That may be appropriate when the desired allocation itself is restricted, but it is not equivalent to placing a floor on the final assignment probabilities. Another tempting alternative is to replace each raw probability by \(\max(r_i,\varepsilon)\) and then renormalize. That is generally unsafe: renormalization can reduce a probability that had just been raised to \(\varepsilon\), violating the intended floor. The floor-and-rescale rule instead pins the affected arms and distributes exactly the remaining probability among the others.

Diagnostics and Limitations

A practical implementation should check the floor condition and the resulting vector at every update. The following checks catch common errors:

• Verify \(0\leq\varepsilon\leq1/K\). A larger common floor is infeasible.
• Check that every final \(p_i\geq\varepsilon\), and that the unrounded probabilities sum to one within numerical tolerance.
• Check that each unpinned arm has \(c r_i\geq\varepsilon\), and each pinned arm has \(c r_i\leq\varepsilon\).
• Handle \(\varepsilon=1/K\) explicitly by returning the uniform vector, not by evaluating a rescaling formula with an empty set of unpinned arms.
• Recompute from the current raw probabilities after each adaptive update; a pinned set from one step need not remain the same at the next step.

A floor changes the randomization probabilities whenever it binds, so it can reduce the degree to which the immediate assignment follows the raw DBCD rule. A large floor forces probabilities toward a more even allocation; at the feasible endpoint it forces uniform randomization regardless of the target or raw probabilities. A small floor may leave the rule unchanged at many steps. The floor does not guarantee a particular realized number of participants on each arm, because assignments remain random, and it does not by itself ensure accurate response estimates or make an unsuitable target appropriate.

Common mistake: “The target share of arm 2 is below \(0.10\), so I should replace that target by \(0.10\) and renormalize the targets.” This changes the target allocation and is not the assignment-probability floor defined here. The correction is to first calculate the raw DBCD probability vector from the specified target and current allocation, then apply the floor-and-rescale transformation to that vector. The target changes only if the design separately specifies a constrained target.